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Steklov regularization and trajectory methods for univariate global optimization

Author

Listed:
  • Orhan Arıkan

    (Bilkent University)

  • Regina S. Burachik

    (University of South Australia)

  • C. Yalçın Kaya

    (University of South Australia)

Abstract

We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions. First, we show that the Steklov regularization convexifies a given univariate coercive function. Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function. For monic quartic polynomials, we prove that this trajectory does generate a global minimizer. In the process, we derive some properties of quartic polynomials. Comparisons are made with a previous approach which uses a quadratic regularization function. We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function.

Suggested Citation

  • Orhan Arıkan & Regina S. Burachik & C. Yalçın Kaya, 2020. "Steklov regularization and trajectory methods for univariate global optimization," Journal of Global Optimization, Springer, vol. 76(1), pages 91-120, January.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00837-3
    DOI: 10.1007/s10898-019-00837-3
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    References listed on IDEAS

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    1. Orhan Arıkan & Regina S. Burachik & C. Yalçın Kaya, 2015. "“Backward Differential Flow” May Not Converge to a Global Minimizer of Polynomials," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 401-408, October.
    2. Jinghao Zhu & Shangrui Zhao & Guohua Liu, 2014. "Solution to Global Minimization of Polynomials by Backward Differential Flow," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 828-836, June.
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    Cited by:

    1. Mikhail Posypkin & Oleg Khamisov, 2021. "Automatic Convexity Deduction for Efficient Function’s Range Bounding," Mathematics, MDPI, vol. 9(2), pages 1-16, January.

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